http://www.ck12.org Chapter 1. Units and Problem Solving Version 2
1.2 Units Concepts
- Every answer to a physics problem must include units. Even if a problem explicitly asks for a speed in meters
per second (m/s), the answer is 5 m/s, not 5. - When you’re not sure how to approach a problem, you can often get insight by considering how to obtain
the units of the desired result by combining the units of the given variables. For instance, if you are given a
distance (in meters) and a time (in hours), the only way to obtain units of speed (meters/hour) is to divide the
distance by the time. This is a simple example of a method calleddimensional analysis, which can be used to
find equations that govern various physical situations without any knowledge of the phenomena themselves.
To use dimensional analysis, assume that the answer to a problem consists of a product of all the variables
given raised to various powers. Many times, there will be only one such combination that gives the desired
result. - This textbook usesSI units(La Système International d’Unités), the most modern form of the metric system.
- When converting speeds from metric to American units, remember the following rule of thumb: a speed
measured inmi/hris about double the value measured inm/s (i.e.,10 m/s is equal to about 20 MPH).
Remember that the speed itself hasn’t changed, just our representation of the speed in a certain set of units. - If a unit is named after a person, it is capitalized. So you write “10 Newtons,” or “10 N,” but “10 meters,” or
“10 m.”
Scalars
The simplest kind of measurement is a single number, orscalar. Scalars are all one needs to describe temperature,
density, length, and many other phenomena in physics. The mathematics used in the manipulation of scalars –
addition, subtraction, multiplication, and division – come naturally to humans, and, to a large extent, to other animals.
Many mammals have an innate ability to divide a pile of food into relatively equal pieces, to distinguish between
objects of different size, and to perform other tasks that seemingly require intelligence. It would seem crazy to
suggest that the animals are performing mathematical operations based on formal logic, but that is not the point.
Much more likely is the idea that formal mathematics is an extension of our natural abilities. In fact, the way math
has been taught throughout history and across the world – think of your own elementary and middle school classes
- seems to reflect this underlying property of human nature.
Vectors
The first new concept introduced here is that of a vector: a scalar magnitude with a direction. In a sense, we are
almost as good at natural vector manipulation as we are at adding numbers. Consider, for instance, throwing a ball
to a friend standing some distance away. To perform an accurate throw, one has to figure out both where to throw
and how hard. We can represent this concept graphically with an arrow: it has an obvious direction, and its length
can represent the distance the ball will travel in a given time. Such a vector (an arrow between the original and final
location of an object) is called a displacement: